Wednesday, April 2, 2014

NOTES ON FERMAT'S LAST THEOREM

Notes on Fermat's Last Theorem

The Fermat equation is a
Diophantine equation which we may try to solve for Z as I did in FLT65. As we
saw before, when p = 3, we have a quadratic equation which we may solve using
the quadratic formula, and we can show relatively easily that the solutions are
non-integer, proving FLT for p = 3. Also I should mention that proof for the
Fermat equation when n = 4, the only non-prime that cannot be reduced to a
prime ≥ 3, was not included
in FLT65 because, as you know, even prior to 1965, mathematicians had proved
that there are no integer solutions for exponents up to a large number,
certainly past nine, which is all that is necessary for application of FLT to
the work Vernon Neppe and I are doing in TDVP.

So, just for the record, since FLT
was proved for all cases from n = 3 to a very large number before Wiles’ proof
in 1994, and even for n = 3 and 4 much before 1965, the question of the
validity of my FLT65 proof has no bearing on the validity of TDVP, which is
supported by some original mathematics developed by me, including an important
application of FLT to elementary particle combination.

FLT states that there are no
integer solutions for any p ≥ 3. If any of the solutions to the p – 1 degree Diophantine
equation (1) are integers, FLT is falsified. If we could solve this Diophantine
equation generalized for all p, we would see that none of the p – 1 solutions
is an integer. But this is not a trivial problem. Number ten on David Hilbert’s
famous list of important unsolved mathematical problems, published in 1900, was
to find a general algorithm for solving Diophantine equations.

A few other historical notes might
be of interest here: The great German mathematician Johann Carl Friedrich Gauss
attempted to solve this problem 100 years before Hilbert, around 1800. To
Gauss, FLT was just a possible conquest as a subset of Diophantine equations in
the larger quest for more general ways to solve Diophantine equations. As part
of this effort, he developed what is now known in number theory as modular algebra.
Appearing almost incomprehensible and complex to the uninitiated, it is really
quite simple: Developed specifically for application to Diophantine equations,
and thus applying only to integers, it states that an integer A is said to be
“congruent” to another integer, B, modulo N, when the difference of the two
integers contains N as a factor. It is
written A ≡ B(Mod N), For
example, 5 ≡ 3(Mod 2) simply
means 5 – 3 is divisible by 2. In case you might wonder why the identity symbol
is used here, recall that it is used in geometry to indicate the congruency of
angles, triangles and other geometric forms.

Why would Gauss, or anyone for
that matter, want to express this simple mathematical relationship in such a
convoluted way? The answer lies in the fact that trinomial Diophantine
equations can be reduced to binomials: For example, notice that the Pythagorean
Theorem equation X2 + Y2 = Z2 may be written Z2
– X2 = Y2 and factored to obtain (Z – X)(Z + X) = Y2.
For the Pythagorean integer triple (Diophantine solution) X,Y,Z = 3,4,5: (Z –
X) ≡ Y(Mod Y), and
(Z2 – X2) ≡ Y(Mod Y2), i.e. (25 – 9) ≡ (Mod 16). Gauss saw that this method could also be
applied to the Fermat equation because it is in the same family of trinomial Diophantine
polynomials as the Pythagorean Theorem equation.

By applying modular algebra to the
first few prime integer exponents of Fermat’s equation, one can easily see why
Gauss made the statement he did, when around 1805, he said:

“I confess that
Fermat's Theorem as an isolated proposition has very little interest for me,
because I could easily lay down a multitude of such propositions, which one
could neither prove nor dispose of.”

The problem with modular algebra
as an approach to solving Diophantine equations is that it only applies to
binomial expressions or expressions whose terms can be re-grouped as products of
binomials. Proofs that there are no integer solutions for the Fermat equation
when n = 3 and 4 using this method are relatively easy to obtain because the
first factor is always a binomial and the second factor can be arranged to be
treated as two binomials for n = 3, and 4 binomials when n = 4. But, when n ≥ 5 the task becomes increasingly more
difficult, because the number of possible combinations of terms in the second
factor increases exponentially from one prime to the next, so the task of
checking all of them becomes onerous very quickly. The number of terms in the
second factor of the equation for any prime, p, is equal to 2p -1, i.e. 16 for p = 5, 64 for p = 7, 1,024 for p = 11, etc. And
there appears to be no pattern of combinations that would allow one to check
for integer solutions by groups, just like there is no apparent pattern in the
occurrence of primes.Gauss went on to prove FLT for n =
3 and 4, using what are called Gaussian numbers, complex numbers which when
squared are integers, also known as complex conjugates: a + bi, which
are, of course, binomials handily subject to modular algebra.

A number of professional
mathematicians worked on the problem of trying to find a general algorithm for
solving Diophantine equations until 1970, when Yuri Matiyasevich proved that
that no such general algorithm is possible within the logic of mathematics as
we know it. So there is no known straight-forward way to solve Diophantine
equations for all p. I can tell you, however, that application of the calculus
of dimensional distinctions (CoD), developed from G. Spencer Brown’s Laws of
Form by yours truly, while it does not yield the numerical values of solutions,
does indicate that the Fermat equation has no integer solutions. Of course, I really don’t expect you to accept this based on my statement
alone, and I don’t expect you to want to learn a whole new system of
mathematics to see the logic of it for yourself. So I will try to present the
essence of the CoD argument in conventional concepts below.

Calculus of Dimensional Distinctions Visualization

First, think of algebraic polynomials
as distinct geometric forms. Not in the conventional sense, where x, y and z
are plotted on Cartesian coordinates; instead, think of the exponent of a term
as an indicator of its dimensionality. Thus terms like X, AY, (a + b)Z, etc.
are linear, terms like X2, XY, aY2, etc. are planar,
terms like X3, XY2, XYZ, etc. are three dimensional, and
terms with variables raised to the fourth power are four-dimensional, etc. The
value of X, Y, etc. in each case is a multiple of some common unit. In the case
of Diophantine equations, the unit is simply the unitary integer, 1. In this
scheme of things, the degree of a polynomial indicates the dimensionality of
the form it describes. An unrestricted Diophantine expression is, like the ring
of integers, closed but infinite. If we set a polynomial equal to a finite
constant, and limit its dimensionality, it is a closed and finite distinction,
like a bubble.

The Fermat expression, Zp
- Xp = Y1p, where Y1 is an integer
constant and p = 3, is a closed distinction that can be visualized as a
spherical bubble with radius equal to Y1. And f(Z) = Z2 +
XZ + Z2 is a planar form, like an ellipse cut out of a plane. The
other factor, g(Z) = Z – X, can be visualized as a line. Both factors are
polynomials, and their dimensionometric forms, like that of their parent polynomial,
are closed and finite, limited by the value of Y1. The ellipse, with
major axis smaller than the diameter of the sphere and minor axis smaller than
the radius of the sphere, is enclosed within the sphere. The line, of length
equal to or shorter than the minor axis of the ellipse, is enclosed within the
ellipse. Now visualize a three-dimensional integer grid originating from the
center of the sphere. Each node of the 3D grid is located exactly one unit from
each of the six nodes nearest to it. If there are any integer solutions with Y
= Y1, (X,Z) = (X1,Z1), they occur where one
point on the line and two points on the ellipse coincide with three points on
the surface of the sphere, and then only if those three points coincide with
nodes of the integer grid. It is not too hard to see, even in this simple visualization,
that this is highly unlikely, if not an outright impossibility.

We can extend this visualization
to all values of p, at least conceptually, by observing that for any prime, the
factorization is the same, with f(Z) a hyper-ellipsoid of degree p -1 enclosed
within the hyper-sphere of the Fermat Diophantine equation with Y = Y1,
and g(Z) = Z – X is a line enclosed within the ellipsoid. A hyper-dimensional
integer grid is still possible to visualize because it turns out that this
p-dimensional hyperspace is still Euclidean, and the Euclidean theorem for
division still applies!